Oscillation response critical

The response of the system will depend mainly on the damping coefficient f. When f < 1, the system is underdamped and has an oscillatory response. The smaller the value of f, the greater the overshoot. If f = 1, the system is termed critically damped and has no oscillation. A critically damped system provides the fastest approach to the final value without the overshoot of an underdamped system. Finally, if f > 1, the system is overdamped. An overdamped system is similar to a critically damped system, in that the response never overshoots the final value. However, the approach for an overdamped system is much slower and varies depending upon the value of f. These typical responses are illustrated in Figure 3.27. 

There are a number of criteria by which the desired performance of a closedloop system can be spedlied in the time domain. For example, we could specify that the closedloop system be critically damped so that there is no overshoot or oscillation. We must then select the type of controller and set its tuning constants so that it will give, when coupled with the process, the desired closedloop response. Naturally the control specification must be physically attainable. We cannot make a Boeing 747 jumbo jet airplane behave like an F-IS fighter. We cannot... 

The molecular assays in Clk"mAic2As bom fide rhythms with a predominant effect on circadian rhythm amplitude and no more than a modest effect on phase or period. With circadian per and tim enhancers, we observed reduced enhancer activity and a reduced cycling amplitude in a Clk" background, consistent with the role of Clk in regulating these enhancers. Nonetheless, the phase of oscillating bioluminescence is similar to that of wild-type flies. The presence of molecular rhythms contrasts with the absence of detectable behavioural rhythms. We favour the notion that this reflects a level or amplitude reduction below a critical threshold for behavioural rhythmicity. The absence of anticipation of light—dark transitions makes it very unlikely that an effect restricted to the lateral neurons — the absence of the neuropeptide PDF, for example — is primarily responsible for the behavioural phenotypes. This is also because LD behavioural rhythms are largely normal in flies devoid of PDF or the pacemaker lateral neurons (Renn et al 1999). However, we cannot exclude the possibility of selective effects of Clk" on other behaviourally relevant neurons. 

The specific models we will analyse in this section are an isothermal autocatalytic scheme due to Hudson and Rossler (1984), a non-isothermal CSTR in which two exothermic reactions are taking place, and, briefly, an extension of the model of chapter 2, in which autocatalysis and temperature effects contribute together. In the first of these, chaotic behaviour has been designed in much the same way that oscillations were obtained from multiplicity with the heterogeneous catalysis model of 12.5.2. In the second, the analysis is firmly based on the critical Floquet multiplier as described above, and complex periodic and aperiodic responses are observed about a unique (and unstable) stationary state. The third scheme has coexisting multiple stationary states and higher-order periodicities. 

The effect of the value of the damping coefficient f on the response is shown in Fig. 7.28. For (< 1 the response is seen to be oscillatory or underdamped when ( >1 it is sluggish or overdamped and when (= 1 it is said to be critically damped, i.e. the final value is approached with the greatest speed without overshooting the Final value. When f = 0 there is no damping and the system output oscillates continuously with constant amplitude. 

The critical strength F, which is necessary for the LC to collapse and to create a response signal, is given by the external as well as by the internal parameters. Predominant are the dielectric ones(c, d ), since these parameters determine the frequency and amplitude of the internal oscillation. We have been able to show that pulsed and modulated external fields can only shift the instability point towards smaller or higher values of the critical field strength per cycle. Qualitative changes in the overall behaviour of the driven LC system are impossible. 

Self-sustained reaction rate oscillations have been shown to occur in many heterogeneous catalytic systems Cl—8]. By now, several comprehensive review papers have been published which deal with different aspects of the problem [3, 9, 10]. An impressive volume of theoretical work has also been accumulated [3, 9, ll], which tries to discover, understand, and model the underlying principles and causative factors behind the phenomenon of oscillations. Most of the people working in this area seem to believe that intrinsic surface processes and rates rather than the interaction between physical and chemical processes are responsible for this unexpected and interesting behavior. However, the majority of the available experimental literature (with a few exceptions [7, 13]) does not contain any surface data and information which could help us to critically test and further Improve the hypotheses and ideas set forth in the literature to explain this type of behavior. 

The method works well if A 1 or, in the case of unequal intervals or two-dimensional geometries, where there is some critical, largest effective A greatly exceeding unity. It was found 1149] that the method works very well with a single BI step in the case of (2-D) microdisk simulations, where indeed large effective A values result at the disk edge and it is these that are responsible for the oscillations if CN is used. 

Evidence for wall slip was suggested over thirty years ago [9,32,63]. One of the first attempts at a slip mechanism was the performance of a Mooney analysis by Blyler and Hart [32]. Working in the condition of constant pressure, they explicitly pointed out melt slip at or near the wall of the capillary as the cause of flow discontinuity. On the other hand, they continued to insist that bulk elastic properties of the polymer melt are responsible for the flow breakdown on the basis that the critical stress for the flow discontinuity transition was found to be quite insensitive to molecular weight. Lack of an explicit interfacial mechanism for slip prevented Blyler and Hart from generating a satisfactory explanation for the flow oscillation observed under a constant piston speed. 

Response of a proportionally controlled system to a step increase in the set point from T to Tg. In this example the gain is roughly one-half the critical gain and the transient oscillations are well damped. Note the offset from the set-point temperature. 

Figure 3.6 c and d illustrate amplitude and phase responses of oscillators having different damping coefficients. The step response of a sensor is usually determined by the time constant as well as by the typical rise and response times of the system. Figure 3.6 b shows the response of a critical damped system to a steplike change in the input signal 0 The time constant r (as defined for an exponential response), the 10% to 90% rise time t(o.i/o.9) and the 95% response time t(0 95) are marked. 

For any value of Kc > 0 until the critical value there are two complex conjugate roots with negative real parts. They imply that the response of the reactor to an input change will be a decaying oscillation. 

Figure Al.6.25. Modulus squared of the rephasing, (a), and non-rephasing, R-, (b), response functions versus final time t for a near-critically overdamped Brownian oscillator model M f). The time delay between the second and third pulse, T, is varied as follows (a) from top to bottom, 7=0, 20, 40, 60, 80, 100,...

Fig. 9.10. Dynamics of cytosolic in response to external stimulation in the minimal model for Ca oscillations based on Ca -induced Ca release. In the absence of stimulation (jS = 0%), a stable, low steady-state level of cytosolic Ca (Z, sohd line) is estabUshed. Upon increasing the stimulation, osciUations develop with a frequency that rises with the stimulus level (compare graphs with /3 = 30% and /3 = 60%). Above a critical level of stimulation, oscillations disappear and a stable, hi steady-state level of cytosolic Ca is established ()3 - 90%). The curves were generated by numerical integration of eqns (9.1) with Vq = 1 j,M/s, Vj = 7.3 p,M/s, 65 p,M/s, 500 p,M/s, X r - 2 p,M, = 0-9 p.M, K2-I pM, /Cf = 1 s, = 10 s", n - m - 2, p - 4. The oscillatory...

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